Subgroups — C₂³×D₇×ΓL(1,𝔽₈) #72245784

Order 2352.

552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₇×ΓL(1,𝔽₈)	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₇×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₇×ΓL(1,𝔽₈)	C₂²	1	7	yes	complemented
5	C₂×D₇×ΓL(1,𝔽₈)	C₂²	1	28	yes	direct product factor
6	C₂³×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×C₇×D₇	C₂×C₃	1	14	yes	complemented
8	C₂³×C₃×D₇	—	7	1	no	maximal
9	C₂⁴×ΓL(1,𝔽₈)	—	7	1	no	maximal
10	C₂×C₇×ΓL(1,𝔽₈)	C₂³	1	7	yes	complemented
11	D₇×ΓL(1,𝔽₈)	C₂³	1	8	yes	direct product factor
12	C₂²×C₇²	C₂²×C₃	1	7	yes	complemented
13	C₂×C₇×D₇	C₂²×C₃	1	28	yes	complemented
14	C₂³×C₃×C₇	—	7	1	no	—
15	C₂²×C₃×D₇	—	7	14	no	—
16	C₂³×ΓL(1,𝔽₈)	—	7	14	no	—
17	C₂³×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, direct product factor
18	C₇×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic, complemented, Hall
19	C₂⁴×C₇	—	7	1	no	—
20	C₂³×D₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
21	C₂×C₇²	C₂³×C₃	1	7	yes	complemented
22	C₇×D₇	C₂³×C₃	1	8	yes	complemented
23	C₂²×C₃×C₇	—	7	7	no	—
24	C₂×C₃×D₇	—	7	28	no	—
25	C₂²×ΓL(1,𝔽₈)	—	7	28	no	—
26	C₂²×ΓL(1,𝔽₈)	C₂×D₇	1	7	yes	direct product factor
27	C₂³×C₇	—	6	1	no	—
28	C₂³×C₇	—	7	14	no	—
29	C₂³×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₂³×C₇	C₃×D₇	1	1	yes	characteristic, complemented
31	C₂²×D₇	C₂×ΓL(1,𝔽₈)	1	14	yes	direct product factor
32	C₇²	C₂⁴×C₃	1	1	yes	characteristic, complemented, Sylow
33	C₂⁴×C₃	—	49	1	no	Hall
34	C₂×C₃×C₇	—	7	7	no	—
35	C₃×D₇	—	7	8	no	—
36	C₂×ΓL(1,𝔽₈)	—	7	8	no	—
37	C₂×ΓL(1,𝔽₈)	C₂²×D₇	1	7	yes	direct product factor
38	C₂²×C₇	—	6	7	no	—
39	C₂²×C₇	—	7	28	no	—
40	C₂²×C₇	C₂²×ΓL(1,𝔽₈)	1	7	yes	complemented
41	C₂²×C₇	C₂×C₃×D₇	1	7	yes	complemented
42	C₂×D₇	C₂²×ΓL(1,𝔽₈)	1	28	yes	direct product factor
43	C₂³×C₃	—	7	1	no	—
44	C₂³×C₃	—	49	14	no	—
45	C₃×C₇	—	7	1	no	—
46	ΓL(1,𝔽₈)	C₂³×D₇	1	1	yes	characteristic, direct product factor
47	C₂⁴	—	7	1	no	Sylow
48	C₂×C₇	—	6	7	no	—
49	C₂×C₇	—	7	8	no	—
50	C₂×C₇	C₂³×ΓL(1,𝔽₈)	1	7	yes	complemented
51	C₂×C₇	C₂²×C₃×D₇	1	7	yes	complemented
52	D₇	C₂³×ΓL(1,𝔽₈)	1	8	yes	direct product factor
53	C₂²×C₃	—	7	7	no	—
54	C₂²×C₃	—	49	28	no	—
55	C₂³	—	7	14	no	—
56	C₂³	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
57	C₇	—	6	1	no	—
58	C₇	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
59	C₇	C₂³×C₃×D₇	1	1	yes	characteristic, complemented
60	C₂×C₃	—	7	7	no	—
61	C₂×C₃	—	49	8	no	—
62	C₂²	—	7	28	no	—
63	C₂²	C₂×D₇×ΓL(1,𝔽₈)	1	7	yes	direct product factor
64	C₃	—	7	1	no	Sylow
65	C₂	—	7	8	no	—
66	C₂	C₂²×D₇×ΓL(1,𝔽₈)	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₇²	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂³

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×C₇×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂²×D₇×ΓL(1,𝔽₈)	C₂	direct (N × H)
3. C₂³×C₇×D₇	C₃	semidirect
4. C₂²×C₇×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×D₇×ΓL(1,𝔽₈)	C₂²	direct (N × H)
6. C₂³×C₇²	C₂×C₃	semidirect
7. C₂²×C₇×D₇	C₂×C₃	semidirect
10. C₂×C₇×ΓL(1,𝔽₈)	C₂³	semidirect
11. D₇×ΓL(1,𝔽₈)	C₂³	direct (N × H)
12. C₂²×C₇²	C₂²×C₃	semidirect
13. C₂×C₇×D₇	C₂²×C₃	semidirect
17. C₂³×ΓL(1,𝔽₈)	D₇	direct (N × H)
18. C₇×ΓL(1,𝔽₈)	C₂⁴	semidirect
20. C₂³×D₇	ΓL(1,𝔽₈)	direct (N × H)
21. C₂×C₇²	C₂³×C₃	semidirect
22. C₇×D₇	C₂³×C₃	semidirect
26. C₂²×ΓL(1,𝔽₈)	C₂×D₇	direct (N × H)
29. C₂³×C₇	C₂×ΓL(1,𝔽₈)	semidirect
30. C₂³×C₇	C₃×D₇	semidirect
31. C₂²×D₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
32. C₇²	C₂⁴×C₃	semidirect
37. C₂×ΓL(1,𝔽₈)	C₂²×D₇	direct (N × H)
40. C₂²×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
41. C₂²×C₇	C₂×C₃×D₇	semidirect
42. C₂×D₇	C₂²×ΓL(1,𝔽₈)	direct (N × H)
46. ΓL(1,𝔽₈)	C₂³×D₇	direct (N × H)
50. C₂×C₇	C₂³×ΓL(1,𝔽₈)	semidirect
51. C₂×C₇	C₂²×C₃×D₇	semidirect
52. D₇	C₂³×ΓL(1,𝔽₈)	direct (N × H)
56. C₂³	D₇×ΓL(1,𝔽₈)	direct (N × H)
58. C₇	C₂⁴×ΓL(1,𝔽₈)	semidirect
59. C₇	C₂³×C₃×D₇	semidirect
63. C₂²	C₂×D₇×ΓL(1,𝔽₈)	direct (N × H)
66. C₂	C₂²×D₇×ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₇×ΓL(1,𝔽₈)	2
C₂²×D₇×ΓL(1,𝔽₈)	2
C₂³×C₇×D₇	3
C₂³×C₃×D₇	7
C₂⁴×ΓL(1,𝔽₈)	7

Subgroups — C23×D7×ΓL(1,𝔽8) #72245784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₇×ΓL(1,𝔽₈) #72245784